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\documentclass[a4paper,conference]{IEEEtran/IEEEtran} 1 1 \documentclass[a4paper,conference]{IEEEtran/IEEEtran}
\usepackage{graphicx,color,hyperref} 2 2 \usepackage{graphicx,color,hyperref}
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\usepackage{url} 8 8 \usepackage{url}
\usepackage[normalem]{ulem} 9 9 \usepackage[normalem]{ulem}
% correct bad hyphenation here 10 10 % correct bad hyphenation here
\hyphenation{op-tical net-works semi-conduc-tor} 11 11 \hyphenation{op-tical net-works semi-conduc-tor}
\textheight=26cm 12 12 \textheight=26cm
\setlength{\footskip}{30pt} 13 13 \setlength{\footskip}{30pt}
\pagenumbering{gobble} 14 14 \pagenumbering{gobble}
\begin{document} 15 15 \begin{document}
\title{Filter optimization for real time digital processing of radiofrequency signals: application 16 16 \title{Filter optimization for real time digital processing of radiofrequency signals: application
to oscillator metrology} 17 17 to oscillator metrology}
18 18
\author{\IEEEauthorblockN{A. Hugeat\IEEEauthorrefmark{1}\IEEEauthorrefmark{2}, J. Bernard\IEEEauthorrefmark{2}, 19 19 \author{\IEEEauthorblockN{A. Hugeat\IEEEauthorrefmark{1}\IEEEauthorrefmark{2}, J. Bernard\IEEEauthorrefmark{2},
G. Goavec-M\'erou\IEEEauthorrefmark{1}, 20 20 G. Goavec-M\'erou\IEEEauthorrefmark{1},
P.-Y. Bourgeois\IEEEauthorrefmark{1}, J.-M. Friedt\IEEEauthorrefmark{1}} 21 21 P.-Y. Bourgeois\IEEEauthorrefmark{1}, J.-M. Friedt\IEEEauthorrefmark{1}}
\IEEEauthorblockA{\IEEEauthorrefmark{1}FEMTO-ST, Time \& Frequency department, Besan\c con, France } 22 22 \IEEEauthorblockA{\IEEEauthorrefmark{1}FEMTO-ST, Time \& Frequency department, Besan\c con, France }
\IEEEauthorblockA{\IEEEauthorrefmark{2}FEMTO-ST, Computer Science department DISC, Besan\c con, France \\ 23 23 \IEEEauthorblockA{\IEEEauthorrefmark{2}FEMTO-ST, Computer Science department DISC, Besan\c con, France \\
Email: \{pyb2,jmfriedt\}@femto-st.fr} 24 24 Email: \{pyb2,jmfriedt\}@femto-st.fr}
} 25 25 }
\maketitle 26 26 \maketitle
\thispagestyle{plain} 27 27 \thispagestyle{plain}
\pagestyle{plain} 28 28 \pagestyle{plain}
\newtheorem{definition}{Definition} 29 29 \newtheorem{definition}{Definition}
30 30
\begin{abstract} 31 31 \begin{abstract}
Software Defined Radio (SDR) provides stability, flexibility and reconfigurability to 32 32 Software Defined Radio (SDR) provides stability, flexibility and reconfigurability to
radiofrequency signal processing. Applied to oscillator characterization in the context 33 33 radiofrequency signal processing. Applied to oscillator characterization in the context
of ultrastable clocks, stringent filtering requirements are defined by spurious signal or 34 34 of ultrastable clocks, stringent filtering requirements are defined by spurious signal or
noise rejection needs. Since real time radiofrequency processing must be performed in a 35 35 noise rejection needs. Since real time radiofrequency processing must be performed in a
Field Programmable Array to meet timing constraints, we investigate optimization strategies 36 36 Field Programmable Array to meet timing constraints, we investigate optimization strategies
to design filters meeting rejection characteristics while limiting the hardware resources 37 37 to design filters meeting rejection characteristics while limiting the hardware resources
required and keeping timing constraints within the targeted measurement bandwidths. 38 38 required and keeping timing constraints within the targeted measurement bandwidths.
\end{abstract} 39 39 \end{abstract}
40 40
\begin{IEEEkeywords} 41 41 \begin{IEEEkeywords}
Software Defined Radio, Mixed-Integer Linear Programming, Finite Impulse Response filter 42 42 Software Defined Radio, Mixed-Integer Linear Programming, Finite Impulse Response filter
\end{IEEEkeywords} 43 43 \end{IEEEkeywords}
44 44
\section{Digital signal processing of ultrastable clock signals} 45 45 \section{Digital signal processing of ultrastable clock signals}
46 46
Analog oscillator phase noise characteristics are classically performed by downconverting 47 47 Analog oscillator phase noise characteristics are classically performed by downconverting
the radiofrequency signal using a saturated mixer to bring the radiofrequency signal to baseband, 48 48 the radiofrequency signal using a saturated mixer to bring the radiofrequency signal to baseband,
followed by a Fourier analysis of the beat signal to analyze phase fluctuations close to carrier. In 49 49 followed by a Fourier analysis of the beat signal to analyze phase fluctuations close to carrier. In
a fully digital approach, the radiofrequency signal is digitized and numerically downconverted by 50 50 a fully digital approach, the radiofrequency signal is digitized and numerically downconverted by
multiplying the samples with a local numerically controlled oscillator (Fig. \ref{schema}) \cite{rsi}. 51 51 multiplying the samples with a local numerically controlled oscillator (Fig. \ref{schema}) \cite{rsi}.
52 52
\begin{figure}[h!tb] 53 53 \begin{figure}[h!tb]
\begin{center} 54 54 \begin{center}
\includegraphics[width=.8\linewidth]{images/schema} 55 55 \includegraphics[width=.8\linewidth]{images/schema}
\end{center} 56 56 \end{center}
\caption{Fully digital oscillator phase noise characterization: the Device Under Test 57 57 \caption{Fully digital oscillator phase noise characterization: the Device Under Test
(DUT) signal is sampled by the radiofrequency grade Analog to Digital Converter (ADC) and 58 58 (DUT) signal is sampled by the radiofrequency grade Analog to Digital Converter (ADC) and
downconverted by mixing with a Numerically Controlled Oscillator (NCO). Unwanted signals 59 59 downconverted by mixing with a Numerically Controlled Oscillator (NCO). Unwanted signals
and noise aliases are rejected by a Low Pass Filter (LPF) implemented as a cascade of Finite 60 60 and noise aliases are rejected by a Low Pass Filter (LPF) implemented as a cascade of Finite
Impulse Response (FIR) filters. The signal is then decimated before a Fourier analysis displays 61 61 Impulse Response (FIR) filters. The signal is then decimated before a Fourier analysis displays
the spectral characteristics of the phase fluctuations.} 62 62 the spectral characteristics of the phase fluctuations.}
\label{schema} 63 63 \label{schema}
\end{figure} 64 64 \end{figure}
65 65
As with the analog mixer, 66 66 As with the analog mixer,
the non-linear behavior of the downconverter introduces noise or spurious signal aliasing as 67 67 the non-linear behavior of the downconverter introduces noise or spurious signal aliasing as
well as the generation of the frequency sum signal in addition to the frequency difference. 68 68 well as the generation of the frequency sum signal in addition to the frequency difference.
These unwanted spectral characteristics must be rejected before decimating the data stream 69 69 These unwanted spectral characteristics must be rejected before decimating the data stream
for the phase noise spectral characterization. The characteristics introduced between the 70 70 for the phase noise spectral characterization. The characteristics introduced between the
downconverter 71 71 downconverter
and the decimation processing blocks are core characteristics of an oscillator characterization 72 72 and the decimation processing blocks are core characteristics of an oscillator characterization
system, and must reject out-of-band signals below the targeted phase noise -- typically in the 73 73 system, and must reject out-of-band signals below the targeted phase noise -- typically in the
sub -170~dBc/Hz for ultrastable oscillator we aim at characterizing. The filter blocks will 74 74 sub -170~dBc/Hz for ultrastable oscillator we aim at characterizing. The filter blocks will
use most resources of the Field Programmable Gate Array (FPGA) used to process the radiofrequency 75 75 use most resources of the Field Programmable Gate Array (FPGA) used to process the radiofrequency
datastream: optimizing the performance of the filter while reducing the needed resources is 76 76 datastream: optimizing the performance of the filter while reducing the needed resources is
hence tackled in a systematic approach using optimization techniques. Most significantly, we 77 77 hence tackled in a systematic approach using optimization techniques. Most significantly, we
tackle the issue by attempting to cascade multiple Finite Impulse Response (FIR) filters with 78 78 tackle the issue by attempting to cascade multiple Finite Impulse Response (FIR) filters with
tunable number of coefficients and tunable number of bits representing the coefficients and the 79 79 tunable number of coefficients and tunable number of bits representing the coefficients and the
data being processed. 80 80 data being processed.
81 81
\section{Finite impulse response filter} 82 82 \section{Finite impulse response filter}
83 83
We select FIR filter for their unconditional stability and ease of design. A FIR filter is defined 84 84 We select FIR filter for their unconditional stability and ease of design. A FIR filter is defined
by a set of weights $b_k$ applied to the inputs $x_k$ through a convolution to generate the 85 85 by a set of weights $b_k$ applied to the inputs $x_k$ through a convolution to generate the
outputs $y_k$ 86 86 outputs $y_k$
$$y_n=\sum_{k=0}^N b_k x_{n-k}$$ 87 87 $$y_n=\sum_{k=0}^N b_k x_{n-k}$$
88 88
As opposed to an implementation on a general purpose processor in which word size is defined by the 89 89 As opposed to an implementation on a general purpose processor in which word size is defined by the
processor architecture, implementing such a filter on an FPGA offer more degrees of freedom since 90 90 processor architecture, implementing such a filter on an FPGA offer more degrees of freedom since
not only the coefficient values and number of taps must be defined, but also the number of bits 91 91 not only the coefficient values and number of taps must be defined, but also the number of bits
defining the coefficients and the sample size. For this reason, and because we consider pipeline 92 92 defining the coefficients and the sample size. For this reason, and because we consider pipeline
processing (as opposed to First-In, First-Out FIFO memory batch processing) of radiofrequency 93 93 processing (as opposed to First-In, First-Out FIFO memory batch processing) of radiofrequency
signals, High Level Synthesis (HLS) languages \cite{kasbah2008multigrid} are not considered but 94 94 signals, High Level Synthesis (HLS) languages \cite{kasbah2008multigrid} are not considered but
the problem is tackled at the Very-high-speed-integrated-circuit Hardware Description Language (VHDL). 95 95 the problem is tackled at the Very-high-speed-integrated-circuit Hardware Description Language (VHDL).
Since latency is not an issue in a openloop phase noise characterization instrument, the large 96 96 Since latency is not an issue in a openloop phase noise characterization instrument, the large
numbre of taps in the FIR, as opposed to the shorter Infinite Impulse Response (IIR) filter, 97 97 numbre of taps in the FIR, as opposed to the shorter Infinite Impulse Response (IIR) filter,
is not considered as an issue as would be in a closed loop system. 98 98 is not considered as an issue as would be in a closed loop system.
99 99
The coefficients are classically expressed as floating point values. However, this binary 100 100 The coefficients are classically expressed as floating point values. However, this binary
number representation is not efficient for fast arithmetic computation by an FPGA. Instead, 101 101 number representation is not efficient for fast arithmetic computation by an FPGA. Instead,
we select to quantify these floating point values into integer values. This quantization 102 102 we select to quantify these floating point values into integer values. This quantization
will result in some precision loss. 103 103 will result in some precision loss.
104 104
%As illustrated in Fig. \ref{float_vs_int}, we see that we aren't 105 105 %As illustrated in Fig. \ref{float_vs_int}, we see that we aren't
%need too coefficients or too sample size. If we have lot of coefficients but a small sample size, 106 106 %need too coefficients or too sample size. If we have lot of coefficients but a small sample size,
%the first and last are equal to zero. But if we have too sample size for few coefficients that not improve the quality. 107 107 %the first and last are equal to zero. But if we have too sample size for few coefficients that not improve the quality.
108 108
% JMF je ne comprends pas la derniere phrase ci-dessus ni la figure ci dessous 109 109 % JMF je ne comprends pas la derniere phrase ci-dessus ni la figure ci dessous
% AH en gros je voulais dire que prendre trop peu de bit avec trop de coeff, ça induit ta figure (bien mieux faite que moi) 110 110 % AH en gros je voulais dire que prendre trop peu de bit avec trop de coeff, ça induit ta figure (bien mieux faite que moi)
% et que l'inverse trop de bit sur pas assez de coeff on ne gagne rien, je vais essayer de la reformuler 111 111 % et que l'inverse trop de bit sur pas assez de coeff on ne gagne rien, je vais essayer de la reformuler
112 112
%\begin{figure}[h!tb] 113 113 %\begin{figure}[h!tb]
%\includegraphics[width=\linewidth]{images/float-vs-integer.pdf} 114 114 %\includegraphics[width=\linewidth]{images/float-vs-integer.pdf}
%\caption{Impact of the quantization resolution of the coefficients} 115 115 %\caption{Impact of the quantization resolution of the coefficients}
%\label{float_vs_int} 116 116 %\label{float_vs_int}
%\end{figure} 117 117 %\end{figure}
118 118
\begin{figure}[h!tb] 119 119 \begin{figure}[h!tb]
\includegraphics[width=\linewidth]{images/demo_filtre} 120 120 \includegraphics[width=\linewidth]{images/demo_filtre}
\caption{Impact of the quantization resolution of the coefficients: the quantization is 121 121 \caption{Impact of the quantization resolution of the coefficients: the quantization is
set to 6~bits, setting the 30~first and 30~last coefficients out of the initial 128~band-pass 122 122 set to 6~bits, setting the 30~first and 30~last coefficients out of the initial 128~band-pass
filter coefficients to 0.} 123 123 filter coefficients to 0.}
\label{float_vs_int} 124 124 \label{float_vs_int}
\end{figure} 125 125 \end{figure}
126 126
The tradeoff between quantization resolution and number of coefficients when considering 127 127 The tradeoff between quantization resolution and number of coefficients when considering
integer operations is not trivial. As an illustration of the issue related to the 128 128 integer operations is not trivial. As an illustration of the issue related to the
relation between number of fiter taps and quantization, Fig. \ref{float_vs_int} exhibits 129 129 relation between number of fiter taps and quantization, Fig. \ref{float_vs_int} exhibits
a 128-coefficient FIR bandpass filter designed using floating point numbers (blue). Upon 130 130 a 128-coefficient FIR bandpass filter designed using floating point numbers (blue). Upon
quantization on 6~bit integers, 60 of the 128~coefficients in the beginning and end of the 131 131 quantization on 6~bit integers, 60 of the 128~coefficients in the beginning and end of the
taps become null, making the large number of coefficients irrelevant and allowing to save 132 132 taps become null, making the large number of coefficients irrelevant and allowing to save
processing resource by shrinking the filter length. This tradeoff aimed at minimizing resources 133 133 processing resource by shrinking the filter length. This tradeoff aimed at minimizing resources
to reach a given rejection level, or maximizing out of band rejection for a given computational 134 134 to reach a given rejection level, or maximizing out of band rejection for a given computational
resource, will drive the investigation on cascading filters designed with varying tap resolution 135 135 resource, will drive the investigation on cascading filters designed with varying tap resolution
and tap length, as will be shown in the next section. Indeed, our development strategy closely 136 136 and tap length, as will be shown in the next section. Indeed, our development strategy closely
follows the skeleton approach \cite{crookes1998environment, crookes2000design, benkrid2002towards} 137 137 follows the skeleton approach \cite{crookes1998environment, crookes2000design, benkrid2002towards}
in which basic blocks are defined and characterized before being assembled \cite{hide} 138 138 in which basic blocks are defined and characterized before being assembled \cite{hide}
in a complete processing chain. In our case, assembling the filter blocks is a simpler block 139 139 in a complete processing chain. In our case, assembling the filter blocks is a simpler block
combination process since we assume a single value to be processed and a single value to be 140 140 combination process since we assume a single value to be processed and a single value to be
generated at each clock cycle. The FIR filters will not be considered to decimate in the 141 141 generated at each clock cycle. The FIR filters will not be considered to decimate in the
current implementation: the decimation is assumed to be located after the FIR cascade at the 142 142 current implementation: the decimation is assumed to be located after the FIR cascade at the
moment. 143 143 moment.
144 144
\section{Filter optimization} 145 145 \section{Filter optimization}
146 146
A basic approach for implementing the FIR filter is to compute the transfer function of 147 147 A basic approach for implementing the FIR filter is to compute the transfer function of
a monolithic filter: this single filter defines all coefficients with the same resolution 148 148 a monolithic filter: this single filter defines all coefficients with the same resolution
(number of bits) and processes data represented with their own resolution. Meeting the 149 149 (number of bits) and processes data represented with their own resolution. Meeting the
filter shape requires a large number of coefficients, limited by resources of the FPGA since 150 150 filter shape requires a large number of coefficients, limited by resources of the FPGA since
this filter must process data stream at the radiofrequency sampling rate after the mixer. 151 151 this filter must process data stream at the radiofrequency sampling rate after the mixer.
152 152
An optimization problem \cite{leung2004handbook} aims at improving one or many 153 153 An optimization problem \cite{leung2004handbook} aims at improving one or many
performance criteria within a constrained resource environment. Amongst the tools 154 154 performance criteria within a constrained resource environment. Amongst the tools
developed to meet this aim, Mixed-Integer Linear Programming (MILP) provides the framework to 155 155 developed to meet this aim, Mixed-Integer Linear Programming (MILP) provides the framework to
provide a formal definition of the stated problem and search for an optimal use of available 156 156 provide a formal definition of the stated problem and search for an optimal use of available
resources \cite{yu2007design, kodek1980design}. 157 157 resources \cite{yu2007design, kodek1980design}.
158 158
First we need to ensure that our problem is a real optimization problem. When 159 159 First we need to ensure that our problem is a real optimization problem. When
we design a process inside the FPGA we want reach some requirements by example the 160 160 we design a process inside the FPGA we want reach some requirements by example the
throughput, the computation time or the rejection noise... But we some limited 161 161 throughput, the computation time or the rejection noise... But we some limited
resources to design the process like BRAM (high performance RAM), DSP (Digital Signal Processor) 162 162 resources to design the process like BRAM (high performance RAM), DSP (Digital Signal Processor)
or LUT (Look Up Table). Since we want optimize some criteria and we have some 163 163 or LUT (Look Up Table). Since we want optimize some criteria and we have some
limited resources our problem is a classical optimization problem. 164 164 limited resources our problem is a classical optimization problem.
165 165
Specifically the degrees of freedom when addressing the problem of replacing the single monolithic 166 166 Specifically the degrees of freedom when addressing the problem of replacing the single monolithic
FIR with a cascade of optimized filters are the number of coefficients $N_i$ of each filter $i$, 167 167 FIR with a cascade of optimized filters are the number of coefficients $N_i$ of each filter $i$,
the number of bits $C_i$ representing the coefficients and the number of bits $D_i$ representing 168 168 the number of bits $C_i$ representing the coefficients and the number of bits $D_i$ representing
the data fed to the filter. Because each FIR in the chain is fed the output of the previous stage, 169 169 the data fed to the filter. Because each FIR in the chain is fed the output of the previous stage,
the optimization of the complete processing chain within a constrained resource environment is not 170 170 the optimization of the complete processing chain within a constrained resource environment is not
trivial. The resource occupation of a FIR filter is considered as $D_i+C_i \times N_i)$ which is 171 171 trivial. The resource occupation of a FIR filter is considered as $(D_i+C_i) \times N_i$ which is
the number of bits needed in a worst case condition to represent the output of the FIR. Such an 172 172 the number of bits needed in a worst case condition to represent the output of the FIR. Such an
occupied area estimate assumes that the number of gates scales as the number of bits and the number 173 173 occupied area estimate assumes that the number of gates scales as the number of bits and the number
of coefficients, but does not account for the detailed implementation of the hardware. Indeed, 174 174 of coefficients, but does not account for the detailed implementation of the hardware. Indeed,
various FPGA implementations will provide different hardware functionalities, and we shall consider 175 175 various FPGA implementations will provide different hardware functionalities, and we shall consider
at the end of the design a synthesis step using vendor software to assess the validity of the solution 176 176 at the end of the design a synthesis step using vendor software to assess the validity of the solution
found. As an example of the limitation linked to the lack of detailed hardware consideration, Block Random 177 177 found. As an example of the limitation linked to the lack of detailed hardware consideration, Block Random
Access Memory (BRAM) used to store filter coefficients are not shared amongst filters, and multiplications 178 178 Access Memory (BRAM) used to store filter coefficients are not shared amongst filters, and multiplications
are most efficiently implemented by using Digital Signal Processing (DSP) blocks whose input word 179 179 are most efficiently implemented by using Digital Signal Processing (DSP) blocks whose input word
size is finite. DSPs are a scarce resource to be saved in a practical implementation. Keeping a high 180 180 size is finite. DSPs are a scarce resource to be saved in a practical implementation. Keeping a high
abstraction on the resource occupation is nevertheless selected in the following discussion in order 181 181 abstraction on the resource occupation is nevertheless selected in the following discussion in order
to leave enough degrees of freedom in the problem to try and find original solutions: too many 182 182 to leave enough degrees of freedom in the problem to try and find original solutions: too many
constraints in the initial statement of the problem leave little room for finding an optimal solution. 183 183 constraints in the initial statement of the problem leave little room for finding an optimal solution.
184 184
\begin{figure}[h!tb] 185 185 \begin{figure}[h!tb]
\begin{center} 186 186 \begin{center}
\includegraphics[width=.5\linewidth]{schema2} 187 187 \includegraphics[width=.5\linewidth]{schema2}
\caption{Shape of the filter: the bandpass BP is considered to occupy the initial 188 188 \caption{Shape of the filter: the bandpass BP is considered to occupy the initial
40\% of the Nyquist frequency range, the bandstop the last 40\%, allowing 20\% transition 189 189 40\% of the Nyquist frequency range, the bandstop the last 40\%, allowing 20\% transition
width.} 190 190 width.}
\label{rejection-shape} 191 191 \label{rejection-shape}
\end{center} 192 192 \end{center}
\end{figure} 193 193 \end{figure}
194 194
Following these considerations, the model is expressed as: 195 195 Following these considerations, the model is expressed as:
\begin{align} 196 196 \begin{align}
\begin{cases} 197 197 \begin{cases}
\mathcal{R}_i &= \mathcal{F}(N_i, C_i)\\ 198 198 \mathcal{R}_i &= \mathcal{F}(N_i, C_i)\\
\mathcal{A}_i &= N_i * C_i + D_i\\ 199 199 \mathcal{A}_i &= N_i * C_i + D_i\\
\Delta_i &= \Delta _{i-1} + \mathcal{P}_i 200 200 \Delta_i &= \Delta _{i-1} + \mathcal{P}_i
\end{cases} 201 201 \end{cases}
\label{model-FIR} 202 202 \label{model-FIR}
\end{align} 203 203 \end{align}
To explain the system \ref{model-FIR}, $\mathcal{R}_i$ represents the rejection of depending on $N_i$ and $C_i$, $\mathcal{A}$ 204 204 To explain the system \ref{model-FIR}, $\mathcal{R}_i$ represents the rejection of depending on $N_i$ and $C_i$, $\mathcal{A}$
is a theoretical area occupation of the processing block on the FPGA, and $\Delta_i$ is the total rejection for the current stage $i$. 205 205 is a theoretical area occupation of the processing block on the FPGA, and $\Delta_i$ is the total rejection for the current stage $i$.
Since the function $\mathcal{F}$ cannot be explictly expressed, we run simulations to determine the rejection depending 206 206 Since the function $\mathcal{F}$ cannot be explictly expressed, we run simulations to determine the rejection depending
on $N_i$ and $C_i$. However, selecting the right filter requires a clear definition of the rejection criterion. Selecting an 207 207 on $N_i$ and $C_i$. However, selecting the right filter requires a clear definition of the rejection criterion. Selecting an
incorrect criterion will lead the linear program solver to produce a solution which might not meet the user requirements. 208 208 incorrect criterion will lead the linear program solver to produce a solution which might not meet the user requirements.
Hence, amongst various criteria including the mean or median value of the FIR response in the stopband, we have designed 209 209 Hence, amongst various criteria including the mean or median value of the FIR response in the stopband, we have designed
a criterion aimed at avoiding ripples on passband and considering the maximum of the FIR spectral response in the stopband 210 210 a criterion aimed at avoiding ripples on passband and considering the maximum of the FIR spectral response in the stopband
(Fig. \ref{rejection-shape}). The bandpass criterion is defined as the sum of the absolute values of the spectral response 211 211 (Fig. \ref{rejection-shape}). The bandpass criterion is defined as the sum of the absolute values of the spectral response
in the bandpass, reminiscent of a standard deviation of the spectral response: this criterion must be minimized to avoid 212 212 in the bandpass, reminiscent of a standard deviation of the spectral response: this criterion must be minimized to avoid
ripples in the passband. The stopband transfer function maximum must also be minimized in order to improve the filter 213 213 ripples in the passband. The stopband transfer function maximum must also be minimized in order to improve the filter
rejection capability. Weighing these two criteria allows designing the linear program to be solved. 214 214 rejection capability. Weighing these two criteria allows designing the linear program to be solved.
215 215
\begin{figure}[h!tb] 216 216 \begin{figure}[h!tb]
\includegraphics[width=\linewidth]{images/noise-rejection.pdf} 217 217 \includegraphics[width=\linewidth]{images/noise-rejection.pdf}
\caption{Rejection as a function of number of coefficients and number of bits} 218 218 \caption{Rejection as a function of number of coefficients and number of bits}
\label{noise-rejection} 219 219 \label{noise-rejection}
\end{figure} 220 220 \end{figure}
221 221
The objective function maximizes the noise rejection ($\max(\Delta_{i_{\max}})$) while keeping resource occupation below 222 222 The objective function maximizes the noise rejection ($\max(\Delta_{i_{\max}})$) while keeping resource occupation below
a user-defined threshold. The MILP solver is allowed to choose the number of successive 223 223 a user-defined threshold. The MILP solver is allowed to choose the number of successive
filters, within an upper bound. The last problem is to model the noise rejection. Since filter 224 224 filters, within an upper bound. The last problem is to model the noise rejection. Since filter
noise rejection capability is not modeled with linear equations, a look-up-table is generated 225 225 noise rejection capability is not modeled with linear equations, a look-up-table is generated
for multiple filter configurations in which the $C_i$, $D_i$ and $N_i$ parameters are varied: for each 226 226 for multiple filter configurations in which the $C_i$, $D_i$ and $N_i$ parameters are varied: for each
one of these conditions, the low-pass filter rejection defined as the mean power between 227 227 one of these conditions, the low-pass filter rejection defined as the mean power between
half the Nyquist frequency and the Nyquist frequency is stored as computed by the frequency response 228 228 half the Nyquist frequency and the Nyquist frequency is stored as computed by the frequency response
of the digital filter (Fig. \ref{noise-rejection}). 229 229 of the digital filter (Fig. \ref{noise-rejection}).
230 230
Linear program formalism for solving the problem is well documented: an objective function is 231 231 Linear program formalism for solving the problem is well documented: an objective function is
defined which is linearly dependent on the parameters to be optimized. Constraints are expressed 232 232 defined which is linearly dependent on the parameters to be optimized. Constraints are expressed
as linear equation and solved using one of the available solvers, in our case GLPK\cite{glpk}. 233 233 as linear equation and solved using one of the available solvers, in our case GLPK\cite{glpk}.
With the notation explain in system \ref{model-FIR}, we have defined our linear problem like this: 234 234 With the notation explain in system \ref{model-FIR}, we have defined our linear problem like this:
\paragraph{Variables} 235 235 \paragraph{Variables}
\begin{align*} 236 236 \begin{align*}
x_{i,j} \in \lbrace 0,1 \rbrace & \text{ $i$ is a given filter} \\ 237 237 x_{i,j} \in \lbrace 0,1 \rbrace & \text{ $i$ is a given filter} \\
& \text{ $j$ is the stage} \\ 238 238 & \text{ $j$ is the stage} \\
& \text{ If $x_{i,j}$ is equal to 1, the filter is selected} \\ 239 239 & \text{ If $x_{i,j}$ is equal to 1, the filter is selected} \\
\end{align*} 240 240 \end{align*}
\paragraph{Constants} 241 241 \paragraph{Constants}
\begin{align*} 242 242 \begin{align*}
\mathcal{F} = \lbrace F_1 ... F_p \rbrace & \text{ All possible filters}\\ 243 243 \mathcal{F} = \lbrace F_1 ... F_p \rbrace & \text{ All possible filters}\\
& \text{ $p$ is the number of different filters} \\ 244 244 & \text{ $p$ is the number of different filters} \\
C(i) & \text{ % Constant to let the 245 245 C(i) & \text{ % Constant to let the
number of coefficients %} \\ & \text{ 246 246 number of coefficients %} \\ & \text{
for filter $i$}\\ 247 247 for filter $i$}\\
\pi_C(i) & \text{ % Constant to let the 248 248 \pi_C(i) & \text{ % Constant to let the
number of bits of %}\\ & \text{ 249 249 number of bits of %}\\ & \text{
each coefficient for filter $i$}\\ 250 250 each coefficient for filter $i$}\\
\mathcal{A}_{\max} & \text{ Total space available inside the FPGA} 251 251 \mathcal{A}_{\max} & \text{ Total space available inside the FPGA}
\end{align*} 252 252 \end{align*}
\paragraph{Constraints} 253 253 \paragraph{Constraints}
\begin{align} 254 254 \begin{align}
1 \leq i \leq p & \nonumber\\ 255 255 1 \leq i \leq p & \nonumber\\
1 \leq j \leq q & \text{ $q$ is the max of filter stage} \nonumber \\ 256 256 1 \leq j \leq q & \text{ $q$ is the max of filter stage} \nonumber \\
\forall j, \mathlarger{\sum_{i}} x_{i,j} = 1 & \text{ At most one filter by stage} \nonumber\\ 257 257 \forall j, \mathlarger{\sum_{i}} x_{i,j} = 1 & \text{ At most one filter by stage} \nonumber\\
\mathcal{S}_0 = 0 & \text{ initial occupation} \nonumber\\ 258 258 \mathcal{S}_0 = 0 & \text{ initial occupation} \nonumber\\
\forall j, \mathcal{S}_j = \mathcal{S}_{j-1} + \forall i, x_{i,j} \times \mathcal{A}_i \label{cstr_size} \\ 259 259 \forall j, \mathcal{S}_j = \mathcal{S}_{j-1} + \mathlarger{\sum_i (x_{i,j} \times \mathcal{A}_i)} \label{cstr_size} \\
\mathcal{S} \leq \mathcal{S}_{\max}\nonumber \\ 260 260 \mathcal{S} \leq \mathcal{S}_{\max}\nonumber \\
\mathcal{N}_0 = 0 & \text{ initial rejection}\nonumber\\ 261 261 \mathcal{N}_0 = 0 & \text{ initial rejection}\nonumber\\
\forall j, \mathcal{N}_j = \mathcal{N}_{j-1} + \forall i, x_{i,j} \times \mathcal{R}_i \label{cstr_rejection} \\ 262 262 \forall j, \mathcal{N}_j = \mathcal{N}_{j-1} + \mathlarger{\sum_i (x_{i,j} \times \mathcal{R}_i)} \label{cstr_rejection} \\
\mathcal{N}_q \geqslant 160 & \text{ an user defined bound}\nonumber\\ 263 263 \mathcal{N}_q \geqslant 160 & \text{ an user defined bound}\nonumber\\
& \text{ (e.g. 160~dB here)}\nonumber\\\nonumber 264 264 & \text{ (e.g. 160~dB here)}\nonumber\\\nonumber
\end{align} 265 265 \end{align}
\paragraph{Goal} 266 266 \paragraph{Goal}
\begin{align*} 267 267 \begin{align*}
\min \mathcal{S}_q 268 268 \min \mathcal{S}_q
\end{align*} 269 269 \end{align*}
270 270
% AH j'aimerai mettre deux equations avec un label mais je ne sais pas comment faire 271 271 % AH j'aimerai mettre deux equations avec un label mais je ne sais pas comment faire
The constraint \ref{cstr_size} means the occupation for the current stage $j$ depends on 272 272 The constraint \ref{cstr_size} means the occupation for the current stage $j$ depends on
the previous occupation and the occupation of current selected filter (it is possible 273 273 the previous occupation and the occupation of current selected filter (it is possible
that no filter is selected for this stage). And the second one \ref{cstr_rejection} 274 274 that no filter is selected for this stage). And the second one \ref{cstr_rejection}
means the same thing but for the rejection, the rejection depends the previous rejection 275 275 means the same thing but for the rejection, the rejection depends the previous rejection
plus the rejection of selected filter. 276 276 plus the rejection of selected filter.
277 277
The MILP solver provides a solution to the problem by selecting a series of small FIR with 278 278 The MILP solver provides a solution to the problem by selecting a series of small FIR with
increasing number of bits representing data and coefficients as well as an increasing number 279 279 increasing number of bits representing data and coefficients as well as an increasing number
of coefficients, instead of a single monolithic filter. Fig. \ref{compare-fir} exhibits the 280 280 of coefficients, instead of a single monolithic filter. Fig. \ref{compare-fir} exhibits the
performance comparison between one solution and a monolithic FIR when selecting a cutoff 281 281 performance comparison between one solution and a monolithic FIR when selecting a cutoff
frequency of half the Nyquist frequency: a series of 5 FIR and a series of 10 FIR with the 282 282 frequency of half the Nyquist frequency: a series of 5 FIR and a series of 10 FIR with the
same space usage are provided as selected by the MILP solver. The FIR cascade provides improved 283 283 same space usage are provided as selected by the MILP solver. The FIR cascade provides improved
rejection than the monolithic FIR at the expense of a lower cutoff frequency which remains to 284 284 rejection than the monolithic FIR at the expense of a lower cutoff frequency which remains to
be tuned or compensated for. 285 285 be tuned or compensated for.
286 286
\begin{figure}[h!tb] 287 287 \begin{figure}[h!tb]
% \includegraphics[width=\linewidth]{images/compare-fir.pdf} 288 288 % \includegraphics[width=\linewidth]{images/compare-fir.pdf}
\includegraphics[width=\linewidth]{images/fir-mono-vs-fir-series-noise-fixe-jmf-light.pdf} 289 289 \includegraphics[width=\linewidth]{images/fir-mono-vs-fir-series-noise-fixe-jmf-light.pdf}
\caption{Comparison of the rejection capability between a series of FIR and a monolithic FIR 290 290 \caption{Comparison of the rejection capability between a series of FIR and a monolithic FIR
with a cutoff frequency set at half the Nyquist frequency.} 291 291 with a cutoff frequency set at half the Nyquist frequency.}
\label{compare-fir} 292 292 \label{compare-fir}
\end{figure} 293 293 \end{figure}
294 294
The resource occupation when synthesizing such FIR on a Xilinx FPGA is summarized as Tab. \ref{t1}. 295 295 The resource occupation when synthesizing such FIR on a Xilinx FPGA is summarized as Tab. \ref{t1}.
296 296
\begin{table}[h!tb] 297 297 \begin{table}[h!tb]
\caption{Resource occupation on a Xilinx Zynq-7000 series FPGA when synthesizing the FIR cascade 298 298 \caption{Resource occupation on a Xilinx Zynq-7000 series FPGA when synthesizing the FIR cascade
identified as optimal by the MILP solver within a finite resource criterion. The last line refers 299 299 identified as optimal by the MILP solver within a finite resource criterion. The last line refers
to available resources on a Zynq-7020 as found on the Zedboard.} 300 300 to available resources on a Zynq-7020 as found on the Zedboard.}
\begin{center} 301 301 \begin{center}
\begin{tabular}{|c|cccc|}\hline 302 302 \begin{tabular}{|c|cccc|}\hline
FIR & BlockRAM & LookUpTables & DSP & rejection (dB)\\\hline\hline 303 303 FIR & BlockRAM & LookUpTables & DSP & rejection (dB)\\\hline\hline
1 (monolithic) & 1 & 76183 & 220 & -162 \\ 304 304 1 (monolithic) & 1 & 76183 & 220 & -162 \\
5 & 5 & 18597 & 220 & -160 \\ 305 305 5 & 5 & 18597 & 220 & -160 \\
10 & 8 & 24729 & 220 & -161 \\\hline\hline 306 306 10 & 8 & 24729 & 220 & -161 \\\hline\hline
\textbf{Zynq 7020} & \textbf{420} & \textbf{53200} & \textbf{220} & \\\hline 307 307 \textbf{Zynq 7020} & \textbf{420} & \textbf{53200} & \textbf{220} & \\\hline
\end{tabular} 308 308 \end{tabular}
\end{center} 309 309 \end{center}
%\vspace{-0.7cm} 310 310 %\vspace{-0.7cm}
\label{t1} 311 311 \label{t1}
\end{table} 312 312 \end{table}
313 313
\section{Filter coefficient selection} 314 314 \section{Filter coefficient selection}
315 315
The coefficients of a single monolithic filter are computed as the impulse response 316 316 The coefficients of a single monolithic filter are computed as the impulse response
of the filter transfer function, and practically approximated by a multitude of methods 317 317 of the filter transfer function, and practically approximated by a multitude of methods
including least square optimization (Matlab's {\tt firls} function), Hamming or Kaiser windowing 318 318 including least square optimization (Matlab's {\tt firls} function), Hamming or Kaiser windowing
(Matlab's {\tt fir1} function). 319 319 (Matlab's {\tt fir1} function).
320 320
\begin{figure}[h!tb] 321 321 \begin{figure}[h!tb]
\includegraphics[width=\linewidth]{images/fir1-vs-firls} 322 322 \includegraphics[width=\linewidth]{images/fir1-vs-firls}
\caption{Evolution of the rejection capability of least-square optimized filters and Hamming 323 323 \caption{Evolution of the rejection capability of least-square optimized filters and Hamming
FIR filters as a function of the number of coefficients, for floating point numbers and 8-bit 324 324 FIR filters as a function of the number of coefficients, for floating point numbers and 8-bit
encoded integers.} 325 325 encoded integers.}
\label{2} 326 326 \label{2}
\end{figure} 327 327 \end{figure}
328 328
Cascading filters opens a new optimization opportunity by 329 329 Cascading filters opens a new optimization opportunity by
selecting various coefficient sets depending on the number of coefficients. Fig. \ref{2} 330 330 selecting various coefficient sets depending on the number of coefficients. Fig. \ref{2}
illustrates that for a number of coefficients ranging from 8 to 47, {\tt fir1} provides a better 331 331 illustrates that for a number of coefficients ranging from 8 to 47, {\tt fir1} provides a better
rejection than {\tt firls}: since the linear solver increases the number of coefficients along 332 332 rejection than {\tt firls}: since the linear solver increases the number of coefficients along
the processing chain, the type of selected filter also changes depending on the number of coefficients 333 333 the processing chain, the type of selected filter also changes depending on the number of coefficients
and evolves along the processing chain. 334 334 and evolves along the processing chain.
335 335
\section{Conclusion} 336 336 \section{Conclusion}
337 337
We address the optimization problem of designing a low-pass filter chain in a Field Programmable Gate 338 338 We address the optimization problem of designing a low-pass filter chain in a Field Programmable Gate
Array for improved noise rejection within constrained resource occupation, as needed for 339 339 Array for improved noise rejection within constrained resource occupation, as needed for
real time processing of radiofrequency signal when characterizing spectral phase noise 340 340 real time processing of radiofrequency signal when characterizing spectral phase noise
characteristics of stable oscillators. The flexibility of the digital approach makes the result 341 341 characteristics of stable oscillators. The flexibility of the digital approach makes the result
best suited for closing the loop and using the measurement output in a feedback loop for 342 342 best suited for closing the loop and using the measurement output in a feedback loop for
controlling clocks, e.g. in a quartz-stabilized high performance clock whose long term behavior 343 343 controlling clocks, e.g. in a quartz-stabilized high performance clock whose long term behavior
is controlled by non-piezoelectric resonator (sapphire resonator, microwave or optical 344 344 is controlled by non-piezoelectric resonator (sapphire resonator, microwave or optical
atomic transition). 345 345 atomic transition).
346 346
\section*{Acknowledgement} 347 347 \section*{Acknowledgement}
348 348
This work is supported by the ANR Programme d'Investissement d'Avenir in 349 349 This work is supported by the ANR Programme d'Investissement d'Avenir in
progress at the Time and Frequency Departments of the FEMTO-ST Institute 350 350 progress at the Time and Frequency Departments of the FEMTO-ST Institute
(Oscillator IMP, First-TF and Refimeve+), and by R\'egion de Franche-Comt\'e. 351 351 (Oscillator IMP, First-TF and Refimeve+), and by R\'egion de Franche-Comt\'e.
The authors would like to thank E. Rubiola, F. Vernotte, G. Cabodevila for support and 352 352 The authors would like to thank E. Rubiola, F. Vernotte, G. Cabodevila for support and
fruitful discussions. 353 353 fruitful discussions.
354 354
\bibliographystyle{IEEEtran} 355 355 \bibliographystyle{IEEEtran}
\bibliography{references,biblio} 356 356 \bibliography{references,biblio}
\end{document} 357 357 \end{document}
358 358
\section{Contexte d'ordonnancement} 359 359 \section{Contexte d'ordonnancement}
Dans cette partie, nous donnerons des d\'efinitions de termes rattach\'es au domaine de l'ordonnancement 360 360 Dans cette partie, nous donnerons des d\'efinitions de termes rattach\'es au domaine de l'ordonnancement
et nous verrons que le sujet trait\'e se rapproche beaucoup d'un problème d'ordonnancement. De ce fait 361 361 et nous verrons que le sujet trait\'e se rapproche beaucoup d'un problème d'ordonnancement. De ce fait
nous pourrons aller plus loin que les travaux vus pr\'ec\'edemment et nous tenterons des approches d'ordonnancement 362 362 nous pourrons aller plus loin que les travaux vus pr\'ec\'edemment et nous tenterons des approches d'ordonnancement
et d'optimisation. 363 363 et d'optimisation.
364 364
\subsection{D\'efinition du vocabulaire} 365 365 \subsection{D\'efinition du vocabulaire}
Avant tout, il faut d\'efinir ce qu'est un problème d'optimisation. Il y a deux d\'efinitions 366 366 Avant tout, il faut d\'efinir ce qu'est un problème d'optimisation. Il y a deux d\'efinitions
importantes à donner. La première est propos\'ee par Legrand et Robert dans leur livre \cite{def1-ordo} : 367 367 importantes à donner. La première est propos\'ee par Legrand et Robert dans leur livre \cite{def1-ordo} :
\begin{definition} 368 368 \begin{definition}
\label{def-ordo1} 369 369 \label{def-ordo1}
Un ordonnancement d'un système de t\^aches $G\ =\ (V,\ E,\ w)$ est une fonction $\sigma$ : 370 370 Un ordonnancement d'un système de t\^aches $G\ =\ (V,\ E,\ w)$ est une fonction $\sigma$ :
$V \rightarrow \mathbb{N}$ telle que $\sigma(u) + w(u) \leq \sigma(v)$ pour toute arête $(u,\ v) \in E$. 371 371 $V \rightarrow \mathbb{N}$ telle que $\sigma(u) + w(u) \leq \sigma(v)$ pour toute arête $(u,\ v) \in E$.
\end{definition} 372 372 \end{definition}
373 373
Dit plus simplement, l'ensemble $V$ repr\'esente les t\^aches à ex\'ecuter, l'ensemble $E$ repr\'esente les d\'ependances 374 374 Dit plus simplement, l'ensemble $V$ repr\'esente les t\^aches à ex\'ecuter, l'ensemble $E$ repr\'esente les d\'ependances
des t\^aches et $w$ les temps d'ex\'ecution de la t\^ache. La fonction $\sigma$ donne donc l'heure de d\'ebut de 375 375 des t\^aches et $w$ les temps d'ex\'ecution de la t\^ache. La fonction $\sigma$ donne donc l'heure de d\'ebut de
chacune des t\^aches. La d\'efinition dit que si une t\^ache $v$ d\'epend d'une t\^ache $u$ alors 376 376 chacune des t\^aches. La d\'efinition dit que si une t\^ache $v$ d\'epend d'une t\^ache $u$ alors
la date de d\'ebut de $v$ sera plus grande ou \'egale au d\'ebut de l'ex\'ecution de la t\^ache $u$ plus son 377 377 la date de d\'ebut de $v$ sera plus grande ou \'egale au d\'ebut de l'ex\'ecution de la t\^ache $u$ plus son
temps d'ex\'ecution. 378 378 temps d'ex\'ecution.
379 379
Une autre d\'efinition importante qui est propos\'ee par Leung et al. \cite{def2-ordo} est : 380 380 Une autre d\'efinition importante qui est propos\'ee par Leung et al. \cite{def2-ordo} est :
\begin{definition} 381 381 \begin{definition}
\label{def-ordo2} 382 382 \label{def-ordo2}
L'ordonnancement traite de l'allocation de ressources rares à des activit\'es avec 383 383 L'ordonnancement traite de l'allocation de ressources rares à des activit\'es avec
l'objectif d'optimiser un ou plusieurs critères de performance. 384 384 l'objectif d'optimiser un ou plusieurs critères de performance.
\end{definition} 385 385 \end{definition}
386 386
Cette d\'efinition est plus g\'en\'erique mais elle nous int\'eresse d'avantage que la d\'efinition \ref{def-ordo1}. 387 387 Cette d\'efinition est plus g\'en\'erique mais elle nous int\'eresse d'avantage que la d\'efinition \ref{def-ordo1}.
En effet, la partie qui nous int\'eresse dans cette première d\'efinition est le respect de la pr\'ec\'edance des t\^aches. 388 388 En effet, la partie qui nous int\'eresse dans cette première d\'efinition est le respect de la pr\'ec\'edance des t\^aches.
Dans les faits les dates de d\'ebut ne nous int\'eressent pas r\'eellement. 389 389 Dans les faits les dates de d\'ebut ne nous int\'eressent pas r\'eellement.
390 390
En revanche la d\'efinition \ref{def-ordo2} sera au c\oe{}ur du projet. Pour se convaincre de cela, 391 391 En revanche la d\'efinition \ref{def-ordo2} sera au c\oe{}ur du projet. Pour se convaincre de cela,
il nous faut d'abord d\'efinir quel est le type de problème d'ordonnancement qu'on traite et quelles 392 392 il nous faut d'abord d\'efinir quel est le type de problème d'ordonnancement qu'on traite et quelles
sont les m\'ethodes qu'on peut appliquer. 393 393 sont les m\'ethodes qu'on peut appliquer.
394 394
Les problèmes d'ordonnancement peuvent être class\'es en diff\'erentes cat\'egories : 395 395 Les problèmes d'ordonnancement peuvent être class\'es en diff\'erentes cat\'egories :
\begin{itemize} 396 396 \begin{itemize}
\item T\^aches ind\'ependantes : dans cette cat\'egorie de problèmes, les t\^aches sont complètement ind\'ependantes 397 397 \item T\^aches ind\'ependantes : dans cette cat\'egorie de problèmes, les t\^aches sont complètement ind\'ependantes
les unes des autres. Dans notre cas, ce n'est pas le plus adapt\'e. 398 398 les unes des autres. Dans notre cas, ce n'est pas le plus adapt\'e.
\item Graphe de t\^aches : la d\'efinition \ref{def-ordo1} d\'ecrit cette cat\'egorie. La plupart du temps, 399 399 \item Graphe de t\^aches : la d\'efinition \ref{def-ordo1} d\'ecrit cette cat\'egorie. La plupart du temps,
les t\^aches sont repr\'esent\'ees par une DAG. Cette cat\'egorie est très proche de notre cas puisque nous devons \'egalement ex\'ecuter 400 400 les t\^aches sont repr\'esent\'ees par une DAG. Cette cat\'egorie est très proche de notre cas puisque nous devons \'egalement ex\'ecuter
des t\^aches qui ont un certain nombre de d\'ependances. On pourra même dire que dans certain cas, 401 401 des t\^aches qui ont un certain nombre de d\'ependances. On pourra même dire que dans certain cas,
on a des anti-arbres, c'est à dire que nous avons une multitude de t\^aches d'entr\'ees qui convergent vers une 402 402 on a des anti-arbres, c'est à dire que nous avons une multitude de t\^aches d'entr\'ees qui convergent vers une
t\^ache de fin. 403 403 t\^ache de fin.
\item Workflow : cette cat\'egorie est une sous cat\'egorie des graphes de t\^aches dans le sens où 404 404 \item Workflow : cette cat\'egorie est une sous cat\'egorie des graphes de t\^aches dans le sens où
il s'agit d'un graphe de t\^aches r\'ep\'et\'e de nombreuses de fois. C'est exactement ce type de problème 405 405 il s'agit d'un graphe de t\^aches r\'ep\'et\'e de nombreuses de fois. C'est exactement ce type de problème
que nous traitons ici. 406 406 que nous traitons ici.
\end{itemize} 407 407 \end{itemize}
408 408
Bien entendu, cette liste n'est pas exhaustive et il existe de nombreuses autres classifications et sous-classifications 409 409 Bien entendu, cette liste n'est pas exhaustive et il existe de nombreuses autres classifications et sous-classifications
de ces problèmes. Nous n'avons parl\'e ici que des cat\'egories les plus communes. 410 410 de ces problèmes. Nous n'avons parl\'e ici que des cat\'egories les plus communes.
411 411
Un autre point à d\'efinir, est le critère d'optimisation. Il y a là encore un grand nombre de 412 412 Un autre point à d\'efinir, est le critère d'optimisation. Il y a là encore un grand nombre de
critères possibles. Nous allons donc parler des principaux : 413 413 critères possibles. Nous allons donc parler des principaux :
\begin{itemize} 414 414 \begin{itemize}
\item Temps de compl\'etion total (ou Makespan en anglais) : ce critère est l'un des critères d'optimisation 415 415 \item Temps de compl\'etion total (ou Makespan en anglais) : ce critère est l'un des critères d'optimisation
les plus courant. Il s'agit donc de minimiser la date de fin de la dernière t\^ache de l'ensemble des 416 416 les plus courant. Il s'agit donc de minimiser la date de fin de la dernière t\^ache de l'ensemble des
t\^aches à ex\'ecuter. L'enjeu de cette optimisation est donc de trouver l'ordonnancement optimal permettant 417 417 t\^aches à ex\'ecuter. L'enjeu de cette optimisation est donc de trouver l'ordonnancement optimal permettant
la fin d'ex\'ecution au plus tôt. 418 418 la fin d'ex\'ecution au plus tôt.
\item Somme des temps d'ex\'ecution (Flowtime en anglais) : il s'agit de faire la somme des temps d'ex\'ecution de toutes les t\^aches 419 419 \item Somme des temps d'ex\'ecution (Flowtime en anglais) : il s'agit de faire la somme des temps d'ex\'ecution de toutes les t\^aches
et d'optimiser ce r\'esultat. 420 420 et d'optimiser ce r\'esultat.
\item Le d\'ebit : ce critère quant à lui, vise à augmenter au maximum le d\'ebit de traitement des donn\'ees. 421 421 \item Le d\'ebit : ce critère quant à lui, vise à augmenter au maximum le d\'ebit de traitement des donn\'ees.
\end{itemize} 422 422 \end{itemize}
423 423
En plus de cela, on peut avoir besoin de plusieurs critères d'optimisation. Il s'agit dans ce cas d'une optimisation 424 424 En plus de cela, on peut avoir besoin de plusieurs critères d'optimisation. Il s'agit dans ce cas d'une optimisation
multi-critères. Bien entendu, cela complexifie d'autant plus le problème car la solution la plus optimale pour un 425 425 multi-critères. Bien entendu, cela complexifie d'autant plus le problème car la solution la plus optimale pour un
des critères peut être très mauvaise pour un autre critère. De ce cas, il s'agira de trouver une solution qui permet 426 426 des critères peut être très mauvaise pour un autre critère. De ce cas, il s'agira de trouver une solution qui permet
de faire le meilleur compromis entre tous les critères. 427 427 de faire le meilleur compromis entre tous les critères.
428 428
\subsection{Formalisation du problème} 429 429 \subsection{Formalisation du problème}
\label{formalisation} 430 430 \label{formalisation}
Maintenant que nous avons donn\'e le vocabulaire li\'e à l'ordonnancement, nous allons pouvoir essayer caract\'eriser 431 431 Maintenant que nous avons donn\'e le vocabulaire li\'e à l'ordonnancement, nous allons pouvoir essayer caract\'eriser
formellement notre problème. En effet, nous allons reprendre les contraintes \'enonc\'ees dans la sections \ref{def-contraintes} 432 432 formellement notre problème. En effet, nous allons reprendre les contraintes \'enonc\'ees dans la sections \ref{def-contraintes}
et nous essayerons de les formaliser le plus finement possible. 433 433 et nous essayerons de les formaliser le plus finement possible.
434 434
Comme nous l'avons dit, une t\^ache est un bloc de traitement. Chaque t\^ache $i$ dispose d'un ensemble de paramètres 435 435 Comme nous l'avons dit, une t\^ache est un bloc de traitement. Chaque t\^ache $i$ dispose d'un ensemble de paramètres
que nous nommerons $\mathcal{P}_{i}$. Cet ensemble $\mathcal{P}_i$ est propre à chaque t\^ache et il variera d'une 436 436 que nous nommerons $\mathcal{P}_{i}$. Cet ensemble $\mathcal{P}_i$ est propre à chaque t\^ache et il variera d'une
t\^ache à l'autre. Nous reviendrons plus tard sur les paramètres qui peuvent composer cet ensemble. 437 437 t\^ache à l'autre. Nous reviendrons plus tard sur les paramètres qui peuvent composer cet ensemble.
438 438
Outre cet ensemble $\mathcal{P}_i$, chaque t\^ache dispose de paramètres communs : 439 439 Outre cet ensemble $\mathcal{P}_i$, chaque t\^ache dispose de paramètres communs :
\begin{itemize} 440 440 \begin{itemize}
\item Dur\'ee de la t\^ache : Comme nous l'avons dit auparavant, dans le cadre d'un FPGA le temps est compt\'e en nombre de coup d'horloge. 441 441 \item Dur\'ee de la t\^ache : Comme nous l'avons dit auparavant, dans le cadre d'un FPGA le temps est compt\'e en nombre de coup d'horloge.
En outre, les blocs sont toujours sollicit\'es, certains même sont capables de lire et de renvoyer une r\'esultat à chaque coups d'horloge. 442 442 En outre, les blocs sont toujours sollicit\'es, certains même sont capables de lire et de renvoyer une r\'esultat à chaque coups d'horloge.
Donc la dur\'ee d'une t\^ache ne peut être le laps de temps entre l'entr\'ee d'une donn\'ee et la sortie d'une autre. Nous d\'efinirons la 443 443 Donc la dur\'ee d'une t\^ache ne peut être le laps de temps entre l'entr\'ee d'une donn\'ee et la sortie d'une autre. Nous d\'efinirons la
dur\'ee comme le temps de traitement d'une donn\'ee, c'est à dire la diff\'erence de temps entre la date de sortie d'une donn\'ee 444 444 dur\'ee comme le temps de traitement d'une donn\'ee, c'est à dire la diff\'erence de temps entre la date de sortie d'une donn\'ee
et de sa date d'entr\'ee. Nous nommerons cette dur\'ee $\delta_i$. % Je devrais la nomm\'ee w comme dans la def2 445 445 et de sa date d'entr\'ee. Nous nommerons cette dur\'ee $\delta_i$. % Je devrais la nomm\'ee w comme dans la def2
\item La pr\'ecision : La pr\'ecision d'une donn\'ee est le nombre de bits significatifs qu'elle compte. En effet, au fil des traitements 446 446 \item La pr\'ecision : La pr\'ecision d'une donn\'ee est le nombre de bits significatifs qu'elle compte. En effet, au fil des traitements
les pr\'ecisions peuvent varier. On nomme donc la pr\'ecision d'entr\'ee d'une t\^ache $i$ comme $\pi_i^-$ et la pr\'ecision en sortie $\pi_i^+$. 447 447 les pr\'ecisions peuvent varier. On nomme donc la pr\'ecision d'entr\'ee d'une t\^ache $i$ comme $\pi_i^-$ et la pr\'ecision en sortie $\pi_i^+$.
\item La fr\'equence du flux en entr\'ee (ou sortie) : Cette fr\'equence repr\'esente la fr\'equence des donn\'ees qui arrivent (resp. sortent). 448 448 \item La fr\'equence du flux en entr\'ee (ou sortie) : Cette fr\'equence repr\'esente la fr\'equence des donn\'ees qui arrivent (resp. sortent).
Selon les t\^aches, les fr\'equences varieront. En effet, certains blocs ralentissent le flux c'est pourquoi on distingue la fr\'equence du 449 449 Selon les t\^aches, les fr\'equences varieront. En effet, certains blocs ralentissent le flux c'est pourquoi on distingue la fr\'equence du
flux en entr\'ee et la fr\'equence en sortie. Nous nommerons donc la fr\'equence du flux en entr\'ee $f_i^-$ et la fr\'equence en sortie $f_i^+$. 450 450 flux en entr\'ee et la fr\'equence en sortie. Nous nommerons donc la fr\'equence du flux en entr\'ee $f_i^-$ et la fr\'equence en sortie $f_i^+$.
\item La quantit\'e de donn\'ees en entr\'ee (ou en sortie) : Il s'agit de la quantit\'e de donn\'ees que le bloc s'attend à traiter (resp. 451 451 \item La quantit\'e de donn\'ees en entr\'ee (ou en sortie) : Il s'agit de la quantit\'e de donn\'ees que le bloc s'attend à traiter (resp.
est capable de produire). Les t\^aches peuvent avoir à traiter des gros volumes de donn\'ees et n'en ressortir qu'une partie. Cette 452 452 est capable de produire). Les t\^aches peuvent avoir à traiter des gros volumes de donn\'ees et n'en ressortir qu'une partie. Cette
fois encore, il nous faut donc diff\'erencier l'entr\'ee et la sortie. Nous nommerons donc la quantit\'e de donn\'ees entrantes $q_i^-$ 453 453 fois encore, il nous faut donc diff\'erencier l'entr\'ee et la sortie. Nous nommerons donc la quantit\'e de donn\'ees entrantes $q_i^-$
et la quantit\'e de donn\'ees sortantes $q_i^+$ pour une t\^ache $i$. 454 454 et la quantit\'e de donn\'ees sortantes $q_i^+$ pour une t\^ache $i$.
\item Le d\'ebit d'entr\'ee (ou de sortie) : Ce paramètre correspond au d\'ebit de donn\'ees que la t\^ache est capable de traiter ou qu'elle 455 455 \item Le d\'ebit d'entr\'ee (ou de sortie) : Ce paramètre correspond au d\'ebit de donn\'ees que la t\^ache est capable de traiter ou qu'elle
fournit en sortie. Il s'agit simplement de l'expression des deux pr\'ec\'edents paramètres. Nous d\'efinirons donc la d\'ebit entrant de la 456 456 fournit en sortie. Il s'agit simplement de l'expression des deux pr\'ec\'edents paramètres. Nous d\'efinirons donc la d\'ebit entrant de la
t\^ache $i$ comme $d_i^-\ =\ q_i^-\ *\ f_i^-$ et le d\'ebit sortant comme $d_i^+\ =\ q_i^+\ *\ f_i^+$. 457 457 t\^ache $i$ comme $d_i^-\ =\ q_i^-\ *\ f_i^-$ et le d\'ebit sortant comme $d_i^+\ =\ q_i^+\ *\ f_i^+$.
\item La taille de la t\^ache : La taille dans les FPGA \'etant limit\'ee, ce paramètre exprime donc la place qu'occupe la t\^ache au sein du bloc. 458 458 \item La taille de la t\^ache : La taille dans les FPGA \'etant limit\'ee, ce paramètre exprime donc la place qu'occupe la t\^ache au sein du bloc.
Nous nommerons $\mathcal{A}_i$ cette taille. 459 459 Nous nommerons $\mathcal{A}_i$ cette taille.
\item Les pr\'ed\'ecesseurs et successeurs d'une t\^ache : cela nous permet de connaître les t\^aches requises pour pouvoir traiter 460 460 \item Les pr\'ed\'ecesseurs et successeurs d'une t\^ache : cela nous permet de connaître les t\^aches requises pour pouvoir traiter
la t\^ache $i$ ainsi que les t\^aches qui en d\'ependent. Ces ensemble sont not\'es $\Gamma _i ^-$ et $ \Gamma _i ^+$ \\ 461 461 la t\^ache $i$ ainsi que les t\^aches qui en d\'ependent. Ces ensemble sont not\'es $\Gamma _i ^-$ et $ \Gamma _i ^+$ \\
%TODO Est-ce vraiment un paramètre ? 462 462 %TODO Est-ce vraiment un paramètre ?
\end{itemize} 463 463 \end{itemize}
464 464
Ces diff\'erents paramètres communs sont fortement li\'es aux \'el\'ements de $\mathcal{P}_i$. Voici quelques exemples de relations 465 465 Ces diff\'erents paramètres communs sont fortement li\'es aux \'el\'ements de $\mathcal{P}_i$. Voici quelques exemples de relations
que nous avons identifi\'ees : 466 466 que nous avons identifi\'ees :
\begin{itemize} 467 467 \begin{itemize}
\item $ \delta _i ^+ \ = \ \mathcal{F}_{\delta}(\pi_i^-,\ \pi_i^+,\ d_i^-,\ d_i^+,\ \mathcal{P}_i) $ donne le temps d'ex\'ecution 468 468 \item $ \delta _i ^+ \ = \ \mathcal{F}_{\delta}(\pi_i^-,\ \pi_i^+,\ d_i^-,\ d_i^+,\ \mathcal{P}_i) $ donne le temps d'ex\'ecution
de la t\^ache en fonction de la pr\'ecision voulue, du d\'ebit et des paramètres internes. 469 469 de la t\^ache en fonction de la pr\'ecision voulue, du d\'ebit et des paramètres internes.
\item $ \pi _i ^+ \ = \ \mathcal{F}_{p}(\pi_i^-,\ \mathcal{P}_i) $, la fonction $F_p$ donne la pr\'ecision en sortie selon la pr\'ecision de d\'epart 470 470 \item $ \pi _i ^+ \ = \ \mathcal{F}_{p}(\pi_i^-,\ \mathcal{P}_i) $, la fonction $F_p$ donne la pr\'ecision en sortie selon la pr\'ecision de d\'epart
et les paramètres internes de la t\^ache. 471 471 et les paramètres internes de la t\^ache.
\item $d_i^+\ =\ \mathcal{F}_d(d_i^-, \mathcal{P}_i)$, la fonction $F_d$ donne le d\'ebit sortant de la t\^ache en fonction du d\'ebit 472 472 \item $d_i^+\ =\ \mathcal{F}_d(d_i^-, \mathcal{P}_i)$, la fonction $F_d$ donne le d\'ebit sortant de la t\^ache en fonction du d\'ebit
sortant et des variables internes de la t\^ache. 473 473 sortant et des variables internes de la t\^ache.
\item $A_i^+\ =\ \mathcal{F}_A(\pi_i^-,\ \pi_i^+,\ d_i^-,\ d_i^+, \mathcal{P}_i)$ 474 474 \item $A_i^+\ =\ \mathcal{F}_A(\pi_i^-,\ \pi_i^+,\ d_i^-,\ d_i^+, \mathcal{P}_i)$
\end{itemize} 475 475 \end{itemize}
Pour le moment, nous ne sommes pas capables de donner une d\'efinition g\'en\'erale de ces fonctions. Mais en revanche, 476 476 Pour le moment, nous ne sommes pas capables de donner une d\'efinition g\'en\'erale de ces fonctions. Mais en revanche,
sur quelques exemples simples (cf. \ref{def-contraintes}), nous parvenons à donner une \'evaluation de ces fonctions. 477 477 sur quelques exemples simples (cf. \ref{def-contraintes}), nous parvenons à donner une \'evaluation de ces fonctions.
478 478
Maintenant que nous avons donn\'e toutes les notations utiles, nous allons \'enoncer des contraintes relatives à notre problème. Soit 479 479 Maintenant que nous avons donn\'e toutes les notations utiles, nous allons \'enoncer des contraintes relatives à notre problème. Soit
un DGA $G(V,\ E)$, on a pour toutes arêtes $(i, j)\ \in\ E$ les in\'equations suivantes : 480 480 un DGA $G(V,\ E)$, on a pour toutes arêtes $(i, j)\ \in\ E$ les in\'equations suivantes :
481 481
\paragraph{Contrainte de pr\'ecision :} 482 482 \paragraph{Contrainte de pr\'ecision :}
Cette in\'equation traduit la contrainte de pr\'ecision d'une t\^ache à l'autre : 483 483 Cette in\'equation traduit la contrainte de pr\'ecision d'une t\^ache à l'autre :
\begin{align*} 484 484 \begin{align*}
\pi _i ^+ \geq \pi _j ^- 485 485 \pi _i ^+ \geq \pi _j ^-
\end{align*} 486 486 \end{align*}
487 487
\paragraph{Contrainte de d\'ebit :} 488 488 \paragraph{Contrainte de d\'ebit :}
Cette in\'equation traduit la contrainte de d\'ebit d'une t\^ache à l'autre : 489 489 Cette in\'equation traduit la contrainte de d\'ebit d'une t\^ache à l'autre :
\begin{align*} 490 490 \begin{align*}
d _i ^+ = q _j ^- * (f_i + (1 / s_j) ) & \text{ où } s_j \text{ est une valeur positive de temporisation de la t\^ache} 491 491 d _i ^+ = q _j ^- * (f_i + (1 / s_j) ) & \text{ où } s_j \text{ est une valeur positive de temporisation de la t\^ache}
\end{align*} 492 492 \end{align*}
493 493
\paragraph{Contrainte de synchronisation :} 494 494 \paragraph{Contrainte de synchronisation :}
Il s'agit de la contrainte qui impose que si à un moment du traitement, le DAG se s\'epare en plusieurs branches parallèles 495 495 Il s'agit de la contrainte qui impose que si à un moment du traitement, le DAG se s\'epare en plusieurs branches parallèles
et qu'elles se rejoignent plus tard, la somme des latences sur chacune des branches soit la même. 496 496 et qu'elles se rejoignent plus tard, la somme des latences sur chacune des branches soit la même.
Plus formellement, s'il existe plusieurs chemins disjoints, partant de la t\^ache $s$ et allant à la t\^ache de $f$ alors : 497 497 Plus formellement, s'il existe plusieurs chemins disjoints, partant de la t\^ache $s$ et allant à la t\^ache de $f$ alors :
\begin{align*} 498 498 \begin{align*}
\forall \text{ chemin } \mathcal{C}1(s, .., f), 499 499 \forall \text{ chemin } \mathcal{C}1(s, .., f),
\forall \text{ chemin } \mathcal{C}2(s, .., f) 500 500 \forall \text{ chemin } \mathcal{C}2(s, .., f)
\text{ tel que } \mathcal{C}1 \neq \mathcal{C}2 501 501 \text{ tel que } \mathcal{C}1 \neq \mathcal{C}2
\Rightarrow 502 502 \Rightarrow
\sum _{i} ^{i \in \mathcal{C}1} \delta_i = \sum _{i} ^{i \in \mathcal{C}2} \delta_i 503 503 \sum _{i} ^{i \in \mathcal{C}1} \delta_i = \sum _{i} ^{i \in \mathcal{C}2} \delta_i
\end{align*} 504 504 \end{align*}
505 505
\paragraph{Contrainte de place :} 506 506 \paragraph{Contrainte de place :}
Cette in\'equation traduit la contrainte de place dans le FPGA. La taille max de la puce FPGA est nomm\'e $\mathcal{A}_{FPGA}$ : 507 507 Cette in\'equation traduit la contrainte de place dans le FPGA. La taille max de la puce FPGA est nomm\'e $\mathcal{A}_{FPGA}$ :
\begin{align*} 508 508 \begin{align*}
\sum ^{\text{t\^ache } i} \mathcal{A}_i \leq \mathcal{A}_{FPGA} 509 509 \sum ^{\text{t\^ache } i} \mathcal{A}_i \leq \mathcal{A}_{FPGA}
\end{align*} 510 510 \end{align*}
511 511
\subsection{Exemples de mod\'elisation} 512 512 \subsection{Exemples de mod\'elisation}
\label{exemples-modeles} 513 513 \label{exemples-modeles}
Nous allons maintenant prendre quelques blocs de traitement simples afin d'illustrer au mieux notre modèle. 514 514 Nous allons maintenant prendre quelques blocs de traitement simples afin d'illustrer au mieux notre modèle.
Pour tous nos exemple, nous prendrons un d\'ebit en entr\'ee de 200 Mo/s avec une pr\'ecision de 16 bit. 515 515 Pour tous nos exemple, nous prendrons un d\'ebit en entr\'ee de 200 Mo/s avec une pr\'ecision de 16 bit.
516 516
Prenons tout d'abord l'exemple d'un bloc de d\'ecimation. Le but de ce bloc est de ralentir le flux en ne gardant 517 517 Prenons tout d'abord l'exemple d'un bloc de d\'ecimation. Le but de ce bloc est de ralentir le flux en ne gardant
que certaines donn\'ees à intervalle r\'egulier. Cet intervalle est appel\'e le facteur de d\'ecimation, on le notera $N$. 518 518 que certaines donn\'ees à intervalle r\'egulier. Cet intervalle est appel\'e le facteur de d\'ecimation, on le notera $N$.
519 519
Donc d'après notre mod\'elisation : 520 520 Donc d'après notre mod\'elisation :
\begin{itemize} 521 521 \begin{itemize}
\item $N \in \mathcal{P}_i$ 522 522 \item $N \in \mathcal{P}_i$
%TODO N ou 1 ? 523 523 %TODO N ou 1 ?
\item $\delta _i = N\ c.h.$ (coup d'horloge) 524 524 \item $\delta _i = N\ c.h.$ (coup d'horloge)
\item $\pi _i ^+ = \pi _i ^- = 16 bits$ 525 525 \item $\pi _i ^+ = \pi _i ^- = 16 bits$
\item $f _i ^+ = f _i ^-$ 526 526 \item $f _i ^+ = f _i ^-$
\item $q _i ^+ = q _i ^- / N$ 527 527 \item $q _i ^+ = q _i ^- / N$
\item $d _i ^+ = q _i ^- / N / f _i ^-$ 528 528 \item $d _i ^+ = q _i ^- / N / f _i ^-$
\item $\Gamma _i ^+ = \Gamma _i ^- = 1$\\ 529 529 \item $\Gamma _i ^+ = \Gamma _i ^- = 1$\\
%TODO Je ne sais pas trouver la taille... 530 530 %TODO Je ne sais pas trouver la taille...
\end{itemize} 531 531 \end{itemize}